# Fit a Periodically Integrated Autoregressive Model.

### Description

Fit a periodically integrated periodic autoregressive model.

### Usage

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### Arguments

`wts` |
a univariate time series object. |

`detcomp` |
a vector indicating the deterministic components included in the auxiliar regression. See
the corresponding item in |

`p` |
the order of the PAR model. In this version first and second order are considered. |

`initvalues` |
by default, initial values are computed for the non-linear modeal. However, in this version there may be cases in which the estimates do not converge, giving an error message. In this case, a numeric vector with initial values guessed by the user can be included. |

### Details

The following equation is estimated by non-linear least squares

* y_t = α_s y_{t-1} + β_s (y_{t-1} - α_{s-1} y_{t-2}) + ε_t,*

under the restriction *Π_{i=1}^{S} α_i = 1* for *s=1,...,S*, where *S* denotes
the number of seasons. Regressors defined in `detcomp`

can also be included. Obviously, for a first
order PIAR process *β* parameters are equal to zero.

### Value

An object of class `fit.piartsm-class`

containing the estimated coefficients in the restricted
non-linear model, the residuals, and the periodic autoregressive coefficients. On the basis of the
estimated *alpha* parameters, the periodically differenced data are also computed. See
`fit.piartsm-class`

for methods that display this information.

### Author(s)

Javier Lopez-de-Lacalle javlacalle@yahoo.es.

### References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

### See Also

`nls`

, `fit.ar.par`

, and `fit.piartsm-class`

.

### Examples

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